["# Solving (x^2 + 144 = 169): A Complete Guide to Finding (x)", "The equation (x^2 + 144 = 169) is a fundamental algebraic equation commonly encountered in high school mathematics. Whether you're solving it for homework, studying equations, or exploring basic algebra, understanding how to isolate (x) is a critical skill. In this comprehensive guide, we’ll walk through every step of solving (x^2 + 144 = 169), explain why the solution works, and highlight the significance of this equation in real-world math.", "## Understanding the Equation", "The equation given is:", "[
\nx^2 + 144 = 169
\n]", "This is a first-degree equation in (x^2). It states that when (x^2) is increased by 144, the total equals 169. Solving this means finding the value(s) of (x) that satisfy this equality.", "---", "## Step-by-Step Solution", "### Step 1: Isolate the (x^2) term", "To solve for (x), first subtract 144 from both sides of the equation:", "[
\nx^2 + 144 - 144 = 169 - 144
\n]", "[
\nx^2 = 25
\n]", "Now, the equation simplifies to (x^2 = 25), which is much easier to solve.", "---", "### Step 2: Apply the Square Root", "To eliminate the square, take the square root of both sides:", "[
\n\sqrt{x^2} = \pm \sqrt{25}
\n]", "[
\nx = \pm 5
\n]", "---", "## Final Solutions", "The equation (x^2 + 144 = 169) has two real solutions:", "[
\nx = 5 \quad \ ext{or} \quad x = -5
\n]", "Both values return the original equation true:", "- For (x = 5): (5^2 + 144 = 25 + 144 = 169)
\n- For (x = -5): ((-5)^2 + 144 = 25 + 144 = 169)", "---", "## Why This Equation Matters", "Solving (x^2 + 144 = 169) is more than just finding roots—it reinforces foundational algebra concepts including:", "- Equations and Isolation: How to isolate variables using inverse operations.
\n- Square Roots: Understanding that squaring or taking square roots introduces positive and negative solutions.
\n- Real-World Applications: Such equations appear in physics (distance, motion), geometry (diagonals of squares), and engineering where balanced mathematical models are essential.", "---", "## How to Verify the Solutions", "Plugging (x = 5) and (x = -5) back into the original equation confirms validity, ensuring correctness and deepening conceptual understanding.", "---", "## Tips for Solving Similar Equations", "- Always move constants to one side before applying square roots.
\n- Don’t forget to consider both positive and negative square roots.
\n- Check answers—especially when the original equation involves squares.", "---", "## Conclusion", "The equation (x^2 + 144 = 169) is a simple yet powerful example of solving quadratic equations through algebraic manipulation. Mastering this problem helps build confidence in handling more complex algebraic expressions and prepares you for advanced math topics. Remember:", "[
\nx^2 + 144 = 169 \implies x = \pm 5
\n]", "Keep practicing—algebra is the language of countless scientific and technical achievements!", "---", "### Further Reading", "- Learn about quadratic equations and the quadratic formula
\n- Explore applications of Pythagorean theorem involving 3-4-5 triangle geometry
\n- Practice solving equations with binomials and leading coefficients", "---", "Keywords: (x^2 + 144 = 169), solve quadratic equations, algebra practice, solving for (x), fundamental math, quadratic solutions, step-by-step algebra, real-world applications of algebra", "---", "Understanding equations like (x^2 + 144 = 169) builds a strong math foundation—start solving with confidence today!"]
